© Board of Trustees, University of Illinois Center for Process Simulation and Design, University of Illinois Robert B. Haber, Duane D. Johnson, and Jonathan A. Dantzig, NSF DMR Spacetime Discontinuous Galerkin Model for Hyperbolic Problems in Physics & Engineering R. B. Haber 1, J. Erickson 2, M. Garland 2, R. Jerrard 3, J. Sullivan 3 and L. Kale 2 1 Theoretical & Applied Mechanics, 2 Computer Science, 3 Mathematics, University of Illinois Research objective: Scientists and engineers use hyperbolic balance and conservation laws to describe a broad range of physical phenomena, from conservation laws that govern gas dynamics to elastodynamic wave equations. Hyperbolic systems are among the most difficult to simulate numerically, especially when shocks are present. We seek improved methods for multi- scale and multi-physics problems. Approach: We are developing adaptive spacetime discontinuous Galerkin (SDG) methods for hyperbolic problems. This includes adaptive spacetime meshing algorithms as well as visualization techniques for data sets on unstructured spacetime grids. Significant results: SDG methods on adaptive, unstructured meshes significantly outperform conventional finite element algorithms for hyperbolic problems and parallelize well. Broader Impact: SDG methods have numerous applications beyond the materials–based problems that motivate our current research. They can be used to advantage in numerous applications, especially those involving moving boundaries and where enforcing local balance is important. Grad students: R. Abedi, Y. Fan, M. Hawker, J. Palaniappan (Theor. & Appl. Mechs.); S.H. Chung, M. Hills, S. Thite, Y. Zhou (Computer Science); K. Jegdic, B. Petracovici (Mathematics) Spacetime discontinuous Galerkin method: Unstructured spacetime grid satisfies a causality condition to guarantee O(N) solution complexity (top left); pixel–accurate rendering of crack- tip wave scattering (center right); adapted space mesh for Mach reflection problem (bottom).

© Board of Trustees, University of Illinois Center for Process Simulation and Design, University of Illinois Robert B. Haber, Duane D. Johnson, and Jonathan A. Dantzig, NSF DMR Spacetime Discontinuous Galerkin Model for Hyperbolic Problems in Physics & Engineering R. B. Haber 1, J. Erickson 2, M. Garland 2, R. Jerrard 3, J. Sullivan 3 and L. Kale 2 1 Theoretical & Applied Mechanics, 2 Computer Science, 3 Mathematics, University of Illinois Research objective: Numerical dissipation in conventional finite element models can easily overwhelm the energetics of phase transitions in shape memory alloys (SMAs) under shock loads. Capturing discontinuities at shocks and moving inter-phase boundaries is an additional challenge. Our goal is to develop high-resolution dynamic models for Austenite–Martensite transitions in SMAs. Approach: We use adaptive spacetime discontinuous Galerkin (SDG) methods to solve a system comprised of the equations of elastodynamics and Hamilton–Jacobi equations that describe phase-boundary dynamics. Significant results: We have demonstrated highly effective adaptive SDG methods for elastic shocks in solids as well as for nonlinear conservation laws (closely related to Hamilton–Jacobi problem). These numerical technologies will be combined to create a coupled model for the target problem. Broader Impact: Dynamic phase transitions in shape memory alloys provide a significant energy absorption mechanism for blast and impact. The SDG modelling capabilities can advance scientific understanding and lead to improved designs for blast- resistant and crashworthy structures. Research Assistants: Reza Abedi, Yong Fan, Jayandran Palaniappan; Dept. of Theoretical & Applied Mechanics Modelling Phase Transitions in Shape Memory Alloys: Austenite-Martensite phase boundary (top; courtesy T. Shield, U. Minn); SDG model of shocks in an elastic solid (middle); adaptive SDG solution of a nonlinear conservation law – inviscid Euler equations for compressible gas dynamics; Mach reflection example, pressure (bottom). We will adapt SDG method for conservation laws to model phase boundary dynamics. Height, color fields are velocity magnitude and strain-energy density.

© Board of Trustees, University of Illinois Center for Process Simulation and Design, University of Illinois Robert B. Haber, Duane D. Johnson, and Jonathan A. Dantzig, NSF DMR Spacetime Discontinuous Galerkin Model for Hyperbolic Problems in Physics & Engineering R. B. Haber 1, J. Erickson 2, M. Garland 2, R. Jerrard 3, J. Sullivan 3 and L. Kale 2 1 Theoretical & Applied Mechanics, 2 Computer Science, 3 Mathematics, University of Illinois Research objective: For multi-scaling, develop method for coupling atomistic and continuum simulations of solids that obeys the full set of mechanics relations, including kinematic compatibility as well as balance of momentum and energy, without ad hoc assumptions. Approach: Using the space-time discontinuous Galerkin (SDG) formalism, we maintain continuity and balance laws across the atomistic/continuum interface via flux-based coupling. The SDG method naturally includes space and time averaging through weak- coupling on boundaries between elements, and thereby transfers coarse-grained information from atomistic to continuum regions. We will employ statistical mechanics methods to repopulate modes on atomistic side that are unresolved on continuum side. Significant Results: (1) Produced a hybrid SDG method that obeys inherently compatibility and energy and momentum balance. (2) Hybrid SDG coarse-grains to continuum and produces exact motion in 1-D. (see figure) (3) Method works with any atomistic potential. For 1-D phonons we use continuum and one embedded atomistic element, we input a traveling wave and plot the wave in both continuum (LHS) and atomic (RHS) regions. Colors show various times. Solution is well represented in all elements. In the SDG Hybrid method, we solve for unknown tractions at atomistic boundary to maintain continuity. Method inherently maintains balance laws. atomisticcontinuum Wave amplitude position Broader Impact: Method potentially allows continuum finite- element calculations of deformation, etc., based on arbitrary atomistic interactions. Offers a consistent method to remove spurious reflections at atomistic-continuum interface.

© Board of Trustees, University of Illinois Center for Process Simulation and Design, University of Illinois Robert B. Haber, Duane D. Johnson, and Jonathan A. Dantzig, NSF DMR Spacetime Discontinuous Galerkin Model for Hyperbolic Problems in Physics & Engineering R. B. Haber 1, J. Erickson 2, M. Garland 2, R. Jerrard 3, J. Sullivan 3 and L. Kale 2 1 Theoretical & Applied Mechanics, 2 Computer Science, 3 Mathematics, University of Illinois Research objective: We seek an effective parallel implementation of a level set method for tracking the solidification front in simulations of solidification processes. A parallel solution strategy is a practical necessity because the simulation problem is computationally complex as it requires a highly refined spatial grid and many time steps to attain the required accuracy. Approach: We use the virtualization concept to implement the parallelization. Significant results: We obtain significant improvement in the parallel performance due to Adaptive overlapping of communication and computation Better cache performance Broader impact: Virtualization might be beneficial in parallelization of other algorithms for moving– boundary problems that require automatic load balancing. A paper is in progress. The Simulation of Solidification Problem Using Level Set Method The relationship of CPU time with respect to the degree of virtualization. 500*500 Grid on 32 Processors500*500 Grid on 16 Processors

© Board of Trustees, University of Illinois Center for Process Simulation and Design, University of Illinois Robert B. Haber, Duane D. Johnson, and Jonathan A. Dantzig, NSF DMR Spacetime Discontinuous Galerkin Model for Hyperbolic Problems in Physics & Engineering R. B. Haber 1, J. Erickson 2, M. Garland 2, R. Jerrard 3, J. Sullivan 3 and L. Kale 2 1 Theoretical & Applied Mechanics, 2 Computer Science, 3 Mathematics, University of Illinois Research objectives: Spacetime discontinuous Galerkin methods nest local finite element solutions within an advancing-front mesh generation algorithm. Thus, our problem reduces to research on efficient patch-wise parallel mesh generation algorithms. Approach: Spacetime methods are set up in such a way that they are easy to parallelize. The solution on each patch depends only on inflow element data. The amount of data per patch is small, making it inexpensive to send patch data to another processor. A Fem framework is used to handle the parallel details at runtime and to partition the space mesh. Significant results: We have demonstrated scalable parallel performance for problems without adaptive mesh refinement on clusters of up to 64 processors with efficiencies as high as 97%. A load-balancing implementation for adaptive models is under development. Broader impact: Adaptive mesh generation for time- dependent problems is an important step in many scientific applications. An efficient parallel algorithm that addresses adaptive mesh generation with an integrated solution procedure has the potential to benefit an extensive set of applications.

© Board of Trustees, University of Illinois Center for Process Simulation and Design, University of Illinois Robert B. Haber, Duane D. Johnson, and Jonathan A. Dantzig, NSF DMR Spacetime Discontinuous Galerkin Model for Hyperbolic Problems in Physics & Engineering R. B. Haber 1, J. Erickson 2, M. Garland 2, R. Jerrard 3, J. Sullivan 3 and L. Kale 2 1 Theoretical & Applied Mechanics, 2 Computer Science, 3 Mathematics, University of Illinois Research Objective: Analysis and design problems characterized by moving interfaces and variable connectivity pose significant computational challenges. Boundary tracking methods require sophisticated meshing technologies and might suffer from mesh tangling and numerical errors associated with remeshing. We seek new methods that circumvent these problems. Approach: We are developing fictitious–domain optimization methods that loosely couple implicit geometry models with finite element response models. We will use this technique to design two- phase microstructures that deliver specified homogenized material properties. An implicit geometry model describes the physical domain boundaries and internal material interfaces. For purposes of response analysis, we project the geometry onto a fictitious- domain described by a fixed finite element grid. Design of microstructure to obtain specified properties

© Board of Trustees, University of Illinois Center for Process Simulation and Design, University of Illinois Robert B. Haber, Duane D. Johnson, and Jonathan A. Dantzig, NSF DMR Spacetime Discontinuous Galerkin Model for Hyperbolic Problems in Physics & Engineering R. B. Haber 1, J. Erickson 2, M. Garland 2, R. Jerrard 3, J. Sullivan 3 and L. Kale 2 1 Theoretical & Applied Mechanics, 2 Computer Science, 3 Mathematics, University of Illinois Significant results: No remeshing is required to accommodate boundary and topological variations. Our method delivers unambiguous optimal geometries that do not require post- processing, and our formulation is suitable for adaptive implementations. Broader Impact: Our topology optimization method is applicable to a broad range of design problems, ranging from large–-scale civil engineering systems to micro and nanoscale systems (e.g., MEMS, microfluidic devices and nanoscale materials design). The numerical techniques are also applicable to modelling microstructure evolution, including topological changes due to nucleation and coarsening.. To illustrate our method, we show results for the shape optimization of a transversely loaded cantilever beam; we minimize compliance subject to a volume constraint. Some of the holes in the initial design coalesce (in a process analogous to coarsening in material microstructures) to produce the optimal design configuration shown.

© Board of Trustees, University of Illinois Center for Process Simulation and Design, University of Illinois Robert B. Haber, Duane D. Johnson, and Jonathan A. Dantzig, NSF DMR Spacetime Discontinuous Galerkin Model for Hyperbolic Problems in Physics & Engineering R. B. Haber 1, J. Erickson 2, M. Garland 2, R. Jerrard 3, J. Sullivan 3 and L. Kale 2 1 Theoretical & Applied Mechanics, 2 Computer Science, 3 Mathematics, University of Illinois Research objective: A wide variety of wavelike physical phenomena are described by systems of hyperbolic partial differential equations. Spacetime discontinuous Galerkin finite element methods developed at UIUC can be used to efficiently compute accurate solutions for many hyperbolic systems, given an appropriate unstructured mesh of the simulation domain in spacetime. We are developing efficient finite-element meshing algorithms to meet the unique requirements of these new numerical methods. Approach: Extending our earlier work on spacetime meshing, we develop a new adaptive advancing-front mesh generation algorithm. Our algorithm adds tetrahedral elements to an evolving unstructured mesh in small patches; the solution within each patch is computed as soon as the patch is created. By responding to numerical error estimates, our algorithm adapts the size and shape of spacetime elements to local geometric and physical features of the solution. Significant results: The meshes generated by our algorithm effectively resolve shocks and other interesting features of the solution, using smaller elements near these features and larger elements everywhere else. This adaptivity allows us to compute numerically accurate solutions several orders of magnitude faster than using a fine mesh everywhere. Linear Elastodynamics: A rectangular plate with a crack is stressed on two opposite sides. The resulting shock wave travels through the plate and reflects off the crack tip. (Only the upper right quadrant of plate is simulated, with symmetric boundary conditions.) The spacetime mesh computed by our algorithm, shown here with time as the vertical axis, accurately captures the passage of shear and pressure waves through the plate. Broader impact: Unlike standard mesh generation in space, meshing directly in spacetime presents unique theoretical challenges that are solved and validated by experiments for linear and nonlinear systems. In concert with new numerical methods, our meshing algorithms promise much more efficient and accurate simulations for a wide variety of physical phenomena of interest to materials scientists and manufacturers. Appeared at 20 th Annual ACM Symposium on Computational Geometry (SoCG), June 2004.

© Board of Trustees, University of Illinois Center for Process Simulation and Design, University of Illinois Robert B. Haber, Duane D. Johnson, and Jonathan A. Dantzig, NSF DMR Spacetime Discontinuous Galerkin Model for Hyperbolic Problems in Physics & Engineering R. B. Haber 1, J. Erickson 2, M. Garland 2, R. Jerrard 3, J. Sullivan 3 and L. Kale 2 1 Theoretical & Applied Mechanics, 2 Computer Science, 3 Mathematics, University of Illinois Research Objectives: Physical simulations (e.g., by finite element methods) typically produce piecewise polynomial solution data, yet have traditionally been displayed using piecewise linear representations. This can result in misleading visualizations that seriously mischaracterize the solution. Approach: We have developed a technique for producing pixel-exact renderings of FEM solutions by exploiting the power of modern programmable graphics processors (GPUs) to directly evaluate solution polynomials in real time. Significant Results: The resulting renderings are of much higher fidelity than those produced by traditional systems. Broader Impact: Engineers performing the simulations can clearly see features of the solution that might otherwise be obscured. Additionally, the lack of artifacts in the rendering makes it far easier to detect errors in the solution itself. Published in IEEE Visualization 2004 with graduate student Y. Zhou, October Above: Spacetime finite element mesh Example shown at left: Time steps from an elastodynamic problem showing crack-tip wave scattering within an elastic solid subjected to shock loading. Both velocity magnitude (height) and strain energy density (color) are shown. Each time step corresponds to a planar slice through the spacetime mesh shown above.

© Board of Trustees, University of Illinois Center for Process Simulation and Design, University of Illinois Robert B. Haber, Duane D. Johnson, and Jonathan A. Dantzig, NSF DMR Spacetime Discontinuous Galerkin Model for Hyperbolic Problems in Physics & Engineering R. B. Haber 1, J. Erickson 2, M. Garland 2, R. Jerrard 3, J. Sullivan 3 and L. Kale 2 1 Theoretical & Applied Mechanics, 2 Computer Science, 3 Mathematics, University of Illinois Research objective: We study the action of shock waves on composite materials with inclusions, as in solid-fuel rocket grains. The need to resolve shock fronts and interfacial damage processes between the matrix and the inclusions makes this a multiscale simulation problem. Numerical simulations predict mechanical response, including shock–induced dewetting of inclusions. Approach: Adaptive spacetime discontinuous Galerkin methods solve multiscale elastodynamics problems; a nonlinear cohesive traction–separation law models the dewetting process. Significant results: In a first high-resolution study of this problem, we observe a complex history of dewetting and rewetting driven by reflections and focusing of shocks between and within the inclusions. Broader impact: These studies provide new insights into the complex behaviour of composite materials under shock loading. These provide a foundation for understanding microstructural damage mechanisms in composite systems and a necessary foundation for modelling detonation in energetic materials. Research Assistants: Reza Abedi, Morgan Hawker; Dept. of Theoretical & Applied Mechanics Research Scientist: Karl Matousz, Center for Simulation of Advanced Rockets Elastodynamic simulation of particle dewetting: An adaptive spacetime discontinuous Galerkin model simulates shock– induced dewetting and rewetting of stiff inclusions. Height and color fields depict velocity magnitude and strain-energy density. Reflections, surface waves and focusing effects within the circular inclusions create a complex history of dewetting and rewetting.